Chapter 4 Analysis Tools Zeinab Eid Algorithm Analysis

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Why we Analyze Algorithms We try to save computer resources such as: § Memory space used to store data structures, Space Complexity § CPU time, which is reflected by Algorithm Run Time Complexity. Zeinab Eid Algorithm Analysis 3

4. 1 Seven Functions 1. The Constant Function: f(n)= c, where c is some fixed constant, c=5, c=100, or c=210, so we let g(n)=1, so f(n)=cg(n) 2. The Logarith Function: f(n)= logbn x = logbn iff bx=n , (see Proposition 4. 1), the most common base for logarithmic function in computer science is 2. 3. Linear Function: f(n)= n , important for algorithms on vectors (one dim. Arrays). Zeinab Eid Algorithm Analysis 4

4. The N-Log-N Function: f(n)= nlogn This function grows a little faster than linear function and a lot slower than quadratic function. 5. The Quadratic Function: f(n)= n 2 It’s important for algorithms of nested loops. 6. The Cubic Function (Polynomial): f(n)= n 3 f(n)=a 0+a 1 n+ a 2 n 2+…+adnd, is a polynomial of degree d, where a 0 , a 1 , …. , ad are constants. 7. The Exponential Function: f(n)= bn Zeinab Eid Algorithm Analysis 5

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We define a set of primitive operations such as the following: • Assigning a value to a variable • Calling a method • Performing an arithmetic operation (for example, adding two numbers) • Comparing two numbers • Indexing into an array • Following an object reference • Returning from a method. Zeinab Eid Algorithm Analysis 15

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Primitive Operations Zeinab Eid Algorithm Analysis 18

Counting Primitive Operations Zeinab Eid Algorithm Analysis 19

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4. 2. 3 Asymptotic Notation Zeinab Eid Algorithm Analysis 23

Example Zeinab Eid Algorithm Analysis 24

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Properties of Big-Oh Notation Zeinab Eid Algorithm Analysis 27

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Big-Omega and Big-Theta Zeinab Eid Algorithm Analysis 34

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Big-Omega • Just as Big-Oh notation provides us a way of saying that a function f(n) is “less than or equal to” another function, • Big-Omega notation provides us a way of saying that a function f(n) is “greater than or equal to” another function. • Let f(n) and g(n) be functions mapping nonnegative integers: We say that f(n) is Ω(g(n)) if g(n) is O(f(n)) • That’s there is a real constant c>0 and an integer constant n 0 ≥ 1 such that: f(n) ≥ cg(n), for n ≥ n 0 Zeinab Eid Algorithm Analysis 36

Big-Theta • In addition, there is a notation that allows us to say that two functions grow at the same rate , up to a constant factor. • We say that f(n) is Θ(g(n)) if f(n) is O(g(n)) and f(n) is Ω(g(n)), • That’s, there are real constants c’>0 and c”>0 and an integer constant n 0 ≥ 1 such that: c’g(n) ≤ f(n) ≥ c”g(n), for n ≥ n 0. Zeinab Eid Algorithm Analysis 37

Example • • f(n) = 3 nlog n + 4 n + 5 log n is Θ(nlog n) Since 3 nlog n + 4 n + 5 log n ≥ 3 nlog n for n ≥ 2 So, f(n) is Ω(nlog n) Since 3 nlog n + 4 n + 5 log n ≤ (3+4+5)nlog n for n ≥ 2 • So, f(n) is O(nlog n) • Thus, f(n) is Θ(nlog n) Zeinab Eid Algorithm Analysis 38

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